Algebra Examples

{cos}^{2} (45)

$\cos^{2} (45)$

Step 1

The exact value of

cos (45)

$\cos (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

{(\frac{\sqrt{2}}{2})}^{2}

${(\frac{\sqrt{2}}{2})}^{2}$

Step 2

Apply the product rule to

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

\frac{{\sqrt{2}}^{2}}{2^{2}}

$\frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 3

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}

$\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 3.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}

$\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 3.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{2^{\frac{2}{2}}}{2^{2}}

$\frac{2^{\frac{2}{2}}}{2^{2}}$

Step 3.4

Cancel the common factor of

2

$2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{2^{2}}$

Step 3.4.2

Rewrite the expression.

$\frac{2^{1}}{2^{2}}$

$\frac{2^{1}}{2^{2}}$

Step 3.5

Evaluate the exponent.

$\frac{2}{2^{2}}$

$\frac{2}{2^{2}}$

Step 4

Raise

$2$ to the power of

$2$ .

$\frac{2}{4}$

Step 5

Cancel the common factor of

$2$ and

$4$ .

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Step 5.1

Factor

$2$ out of

$2$ .

$\frac{2 (1)}{4}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor

$2$ out of

$4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 5.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 5.2.3

Rewrite the expression.

$\frac{1}{2}$

$\frac{1}{2}$

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

$\cos^{2} (45)$

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$